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Full classification of permutation rational functions and complete rational functions of degree three over finite fields

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Abstract

Let q be a prime power, \(\mathbb {F}_q\) be the finite field of order q and \(\mathbb {F}_q(x)\) be the field of rational functions over \(\mathbb {F}_q\). In this paper we classify and count all rational functions \(\varphi \in \mathbb {F}_q(x)\) of degree 3 that induce a permutation of \(\mathbb {P}^1(\mathbb {F}_q)\). As a consequence of our classification, we can show that there is no complete permutation rational function of degree 3 unless \(3\mid q\) and \(\varphi \) is a polynomial.

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Authors and Affiliations

  1. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111, Bonn, Germany

    Andrea Ferraguti

  2. Mathematical Institute, University of Oxford, Woodstock Rd, Oxford, OX2 6GG, UK

    Giacomo Micheli

Authors
  1. Andrea Ferraguti
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  2. Giacomo Micheli
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Correspondence to Giacomo Micheli.

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Communicated by D. Panario.

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The first author was partially supported by the Swiss National Science Foundation Grant Number 168459, and is grateful to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support. The second author was supported by the Swiss National Science Foundation Grant Number 171248.

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Ferraguti, A., Micheli, G. Full classification of permutation rational functions and complete rational functions of degree three over finite fields. Des. Codes Cryptogr. 88, 867–886 (2020). https://doi.org/10.1007/s10623-020-00715-0

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